Place the potatoes into a large pot of salted water. Bring it to a boil and let it simmer for 15 minutes. At this point, they will not be fully cooked.
Take the potatoes out of the water, and allow them to cool for 10 minutes. Slice them into 1/8 inch thick slices.
Pre-heat the broiler. Line a large baking sheet with paper towels. Set aside.
Heat 1-tablespoon olive oil and 1/2 tablespoon butter in a non-stick skillet in medium heat, until sizzling hot. Add in a tablespoon of minced shallot, and cook while stirring it constantly with a wooden spoon, for 30 seconds. Add in 1/4 teaspoon of the minced garlic and cook for 10-15 seconds.
Line the bottom of the skillet with as many potato slices as it can fit and cook 2-3 minutes on each side with the oil, butter and shallot mixture. Once cooked, transfer them to the baking sheet lined with paper towels. Lightly sprinkle them with coarse salt. Discard the oil mixture.
Repeat the same process with the rest of the potato slices: Cook the oils & shallot, sear potatoes, transfer seared potatoes to the baking sheet and sprinkle with salt.
Generously butter the bottom and sides of a 6-inch cast iron skillet.
Line the bottom with a single layer of the potato slices, sprinkle them with black pepper 1/4 teaspoon (or more if you like) chopped thyme. Add another layer until all potato slices are layered. Make sure to press each layer down using the back of a wooden spoon as you go. Pressing them down is a very important step, as doing so is what keeps the cake together.
Place it under the broiler until it is browned and crispy, about 7-9 minutes.
Once baked, allow it to cool for 5 minutes on the kitchen counter. Invert it onto a plate. If you’d like, sprinkle it with more chopped thyme.
Slice it into wedges and serve immediately.